A Finite Automata Reduction Procedure to Improve MIP regular Formulations of Personnel Scheduling Problems
Résumé
Recent work has proposed to model working regulations as formal languages from which network flow formulations can be derived to obtain efficient Mixed Integer Programs (MIP) for Personnel Scheduling Problems (PSP). Such languages can be represented by Finite Automata (FA) whose number of transitions is the number of variables in the flow formulations. Moreover, unambiguity is a property of some FAs which allows to break symmetries in the resulting MIP formulation. To reduce the number of transitions in unambiguous FAs, we adapt an existing FA state reduction procedure. As an intermediate step, our adaptation requires solving an NP-hard variant of the biclique decomposition problem: the Weighted Vertices Biclique Decomposition Problem (WVBDP). We propose and study a regret heuristic to quickly find good solutions to large WVBDPs and be able to use our transition reduction procedure to improve FA-based formulations of practical PSPs. We test our approach on a PSP from the literature. Numerical results show that it efficiently reduces the number of transitions which leads to mathematical models with fewer variables and better solving performances.
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